следить за определённой темой(To keep up to date on the subject, Prince Philip had reportedly maintained a subscription to the journal Flying Saucer Review and had even looked over military reports about the phenomenon.unexplained-mysteries.comART Vancouver)
не отставать от(в смысле "быть в курсе", "следить за") He stays up to date with the latest advancements in dentistry by completing numerous continuing education courses every year. I. Havkin)